# Linear Algebra - Basis

Tags: algebra, basis, linear
 P: 188 Hello... im doing this problem with basis. Infact, im having a lot of problems understanding basis, i did every question in the text book and I still get seem to understand the idea of it. So i was hoping somebody can help me with the whole idea about it. Say, for example, how would i go abouts a question like this: --Let F be a field and let V = F^3. Let W = {(a1 a2 a3) E F^3 / 2a1 - a2 - a3 = 0 } Find a basis for W -- If a question like that came on a test, id fail it - sad to say. It would also be good if somebody knows a good website or has sameple tests that covers this mataril so that i may get used to it. Thanks
 P: 429 so any element in W can be represented like so: w = (a1, a2, a3), where a1, a2, and a3 are arbitrary. but W has the additional restriction that a1 = 1/2 (a2 + a3). so w= ( 1/2 (a2+a3), a2, a3). w = a2 ( 1/2, 1, 0) + a3 (1/2, 0, 1). (it's easy to see that this is the same as above.) so w = span{(1/2, 1, 0), (1/2, 0, 1)}. and that set {(1/2, 1, 0), (1/2, 0, 1)} is our basis. ah, i miss these problems!
 PF Patron Sci Advisor Emeritus P: 11,137 Keep in mind that the above is not the only basis. Also, notice that the given vector space is nothing but a (generalization of a) plane through the origin in $\mathbb{R}^3$. Any pair of vectors in the plane will serve as a basis.
P: 188

## Linear Algebra - Basis

he said that a1 = 1/2 (a2 + a3).

well.. shouldn't it be a1 = 1/2 (a2 - a3) ?

Does it make a difference?
 PF Patron Sci Advisor Emeritus P: 11,137 No, you said "2a1 - a2 -a3 = 0" That gives 2a1 = a2 + a3
 P: 188 oh that was my silly mistake.. but either way.. i still learned something :)
 P: 188 Hey.... what about the subspace... U = (a+b+c=0/ a b c is in the Real Numbers) How would you show the span of that. Also, the comment Gokul43201 made, about the basis being the plane through the origin, how did he know that? I mean its a plane because of the two vectors, but how did he know that its through the origin?
P: 132
 Quote by rad0786 Also, the comment Gokul43201 made, about the basis being the plane through the origin, how did he know that? I mean its a plane because of the two vectors, but how did he know that its through the origin?
The general equation of a plane is $Ax+By+Cz = D$.
$D = 0 \Longleftrightarrow$ the plane goes through the origin.
 P: 188 "and that set {(1/2, 1, 0), (1/2, 0, 1)} is our basis." Can that be a Basis for F^3? what is F (Feild)? isnt that the same as R, like R^3
PF Patron
 Quote by iNCREDiBLE The general equation of a plane is $Ax+By+Cz = D$. $D = 0 \Longleftrightarrow$ the plane goes through the origin.